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A case study on the use of scale separation-based analytic propagators for parameter inference in stochastic gene regulation

01 Jan 2015-Vol. 3, Iss: 2, pp 164-173

AboutThe article was published on 2015-01-01 and is currently open access. It has received 7 citation(s) till now. The article focuses on the topic(s): Inference.

Topics: Inference (55%)

Summary (3 min read)

1. INTRODUCTION

  • Gene expression is a complex and highly regulated multi-step process that is responsible for the timely synthesis of proteins necessary for cellular function.
  • A simple, “two-stage” model for stochastic gene expression consists of a constitutively active gene from which an mRNA molecule can be transcribed, and protein, the production of which depends on the instantaneous abundance of mRNA (see Fig. 1A).
  • In recent work [14], the procedure developed in [2] was extended to capture departure from the assumption of perfect scale separation: the ratio of degradation rates of protein and mRNA, denoted ε, was taken to be small and positive instead of zero, as was the case in [2].
  • While fluorescence microscopy yields only time-series for the intensity, these can nonetheless be converted into absolute protein numbers if a calibration factor of molecules per unit intensity can be estimated, see e.g. [19].
  • For comparison, both propagators are also contrasted with an approximate solution of the CME that is computed using a finite state projection.

2.1. Two-stage Gene Expression Model

  • The authors model gene expression as a two-stage process, whereby DNA is transcribed to mRNA, which is then translated into protein (see Fig. 1A).
  • (1) Here,m and n denote mRNA and protein copy numbers, respectively, a is the non-dimensional transcription rate and b is the non-dimensional translation rate, while the degradation rates of mRNA and protein are given by γ and 1, respectively (cf. Fig. 1A).
  • Finally, τ denotes a suitably nondimensionalized time variable.
  • It follows that for ε sufficiently small, the dynamics of Eq. (1) will vary on two distinct time-scales: the long-term behavior of the system is naturally described on the “slow” τ -scale, while the “fast” transients evolve according to the rescaled time t := τε .

2.2.2. Uniform (First-Order) Propagator

  • Here, ε denotes the per- turbation parameter, as before, while t is the fast time variable.
  • The authors remark that the transition probability Pn|n0(τ, ε) contributes on the slow τ -scale in Eq. (3), while the t-dependent contribution in Eq. (3) accounts for the transient dynamics on the fast time-scale.

2.3. Special Cases of the Hypergeometric Functions

  • Care must be taken when evaluating the hypergeometric function 2F1(a, b, c, z).
  • The following special cases are of use [20].

2.4. Stochastic Simulation

  • Stochastic simulations were performed using the StochKit 2.0 [21] simulation framework and the standard stochastic simulation algorithm [3], with a non-dimensionalized transcription rate a = 20 and a non-dimensionalized translation rate b = 2.5, corresponding to “regime I” as defined in [14].
  • Each value of γ was simulated 20 times, and the resulting trajectories were used for computing the probability landscapes of the rescaled model parameters a and b.
  • Protein quantities were observed without measurement noise at intervals of 0.1 time units.
  • All simulation runs assumed zero molecules of mRNA and protein initially, i.e., m0 = 0 = n0.

2.5. Implementation

  • Both the zeroth-order propagator Pn|n0 , Eq. (2), and the uniform propagator Pn|n0 , Eq. (3), were implemented in C++ with a Matlab mex-file interface.
  • Special functions were evaluated using the GNU scientific library [22], the Hyp_2F1 function implementation of the Gauss hypergeometric function [23], and the Algorithm 910multiprecision special function library [24].
  • While the difference of such numbers is potentially below a double precision machine error of approximately 10−13, they are nonetheless essential in the correct computation of the transition probabilities.
  • Their C++ implementation is still inaccurate in some extreme cases, typically for very large protein numbers n, due to numerical differences which are sometimes as small as 10−370 in Eq. (7), but which unfortunately cannot be neglected as they are inflated by the remaining terms in the expression.
  • Such inaccuracies are infrequent, though, and generally occur during transitions for which the uniform propagator yields nonphysical values; thus, they do not substantially affect their analysis, or the conclusions obtained in this study.

3. RESULTS AND DISCUSSION

  • To assess the applicability of the zeroth-order propagator Pn|n0(τ, 0), Eq. (2), and the uniform propagator Pn|n0(τ, t, ε), Eq. (3), for parameter inference in the two-stage gene expression model, the authors simulated time-series with a specific parameter pair (a∗, b∗).
  • Then, the authors computed the likelihood of the observed data set on the basis of the two propagators for a range of values for the parameters a and b.
  • For simplicity, the authors assumed the scale separation γ between mRNA and protein lifetimes to be known (see Methods for definitions).

3.2. Parameter Inference

  • The term ni represents the number of proteins at measurement time ti.
  • Thus, the authors compute the logarithm of the probability of each transition, from ni−1 protein molecules at time ti−1 to ni molecules at time ti, in the sequence of observed measurements (see Fig. 1B, inset).
  • In order to estimate the parameters a and b from simulated protein time-courses, Eq. (13) has to be evaluated very frequently.
  • The authors thus developed a numerically stable expression for the uniform propagator Pn|n0 (see Section 2 2.2), and they used an efficient implementation in C++ for both propagators that results in reasonable runtimes; see Section 2 2.5 for details.

3.3. Comparison of Propagator Accuracy and Efficiency

  • For each pair (a, b), the authors computed the log-likelihood L(a, b), thus obtaining a likelihood landscape that should ideally have its maximum, the maximum likelihood estimator (MLE), at the true parameter values (a∗, b∗).
  • The authors immediately encountered the obstacle that the uniform propagator Pn|n0 yields negative transition probabilities, or even probabilities larger than one, for some choices of (a, b).
  • The resulting non-physicality is discussed in [14], and is due to the fact that Pn|n0 is derived from an asymptotic approximation; nonetheless, it is problematic when computing the overall log-likelihood, as the definition in Eq. (13) becomes meaningless.
  • In Fig. 3B, a typical protein time-course with γ = 10 is shown (top), along with the log-likelihood obtained from the uniform propagator Pn|n0 , Eq. (3), for the true parameter values , and for the MLE (cyan).
  • From that plot, it is obvious that large regions of the transition space yield non-physical values, shown in gray.

4. CONCLUSION

  • The authors have investigated the utility of a propagator-based approach for approximating the transition probabilities in a simple two-stage gene expression model by attempting parameter inference from protein time-series.
  • The simulations were initialized with zero molecules of both mRNA and protein; this simplification, as compared to a typical biological setting, does not affect the subsequent analysis.
  • Alternatively, one could use Markov Chain Monte Carlo (MCMC) techniques to sample directly from the posterior in order to obtain the log-likelihood landscape [26].
  • The variance of the noise then constitutes an additional unknown parameter σ which would have to be inferred.

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Edinburgh Research Explorer
A case study on the use of scale separation-based analytic
propagators for parameter inference in stochastic gene
regulation
Citation for published version:
Feigelman, J, Popovic, N & Marr, C 2015, 'A case study on the use of scale separation-based analytic
propagators for parameter inference in stochastic gene regulation', Journal of Coupled Systems and
Multiscale Dynamics, vol. 3, no. 2, pp. 164-173. https://doi.org/10.1166/jcsmd.2015.1074
Digital Object Identifier (DOI):
10.1166/jcsmd.2015.1074
Link:
Link to publication record in Edinburgh Research Explorer
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Journal of Coupled Systems and Multiscale Dynamics
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Download date: 09. Aug. 2022

Journal of Coupled Systems and Multiscale Dynamics
A case study on the use of scale separation-based analytical propagators for
parameter inference in models of stochastic gene regulation
Justin Feigelman
1,2
, Nikola Popovi
´
c
3
, and Carsten Marr
1
1
Helmholtz Zentrum München - German Research Center for Environmental Health,
Institute of Computational Biology, Ingolstädter Landstraße 1, 85764 Neuherberg Germany
2
Technische Universität München, Center for Mathematics,
Chair of Mathematical Modeling of Biological Systems,
Boltzmannstraße 3, 85748 Garching, Germany and
3
University of Edinburgh, School of Mathematics and Maxwell Institute for Mathematical Sciences,
James Clerk Maxwell Building, King’s Buildings,
Peter Guthrie Tait Road, Edinburgh EH9 3FD, United Kingdom
(Dated: October 2, 2015)
Advances in long-term fluorescent time-lapse microscopy have made it possible to study the expression of
individual genes in single cells. In a typical setting, the intensity of one or more fluorescently-labeled proteins
is measured at regular time intervals. Such time-courses are inherently noisy due to both measurement noise
and intrinsic stochasticity of the underlying gene expression regulation. Fitting stochastic models to time-series
data remains a difficult task, partly because analytical and tractable expressions for the transition probabilities
cannot usually be derived in closed form.
In the present work, we employ a recently developed approach that is based on geometric singular perturba-
tion theory, as applied to the chemical master equation of a simple two-stage gene expression model, to compute
parameter likelihoods using synthetic protein time-series. We study the identifiability of model parameters in
this simple setting, and compare the performance of the perturbative (uniform) propagator to a previously pub-
lished, idealized (zeroth-order) propagator that assumes perfect time-scale separation between degradation of
mRNA and protein. We find that both propagators are useful for parameter inference when the scale separation is
sufficiently large. However, with decreasing separation, the uniform propagator sometimes yields non-physical
negative transition probabilities which render parameter inference difficult. Finally, we discuss the utility of
both propagators, and possible extensions thereof, for inference. For computational efficiency, the propagators
were implemented in C++ and embedded in Matlab; the code is available upon request.
Keywords: Multi-scale gene expression dynamics. Propagator approximation. Parameter inference. Geometric
singular perturbation.
1. INTRODUCTION
Gene expression is a complex and highly regu-
lated multi-step process that is responsible for the
timely synthesis of proteins necessary for cellular
function. At the molecular level, gene expression is
inherently stochastic due to random binding events
of transcription factors and the transcriptional ma-
chinery, which ultimately leads to mRNA transcrip-
tion with probabilities depending on the concentra-
tion of the reaction educts. Protein synthesis re-
quires a chance encounter of mRNA with ribosomes,
and mRNA or protein degradation an encounter with
the degradation machinery. Thus, models for gene
expression have to capture the stochasticity at both
mRNA and protein levels.
A simple, “two-stage” model for stochastic gene
expression consists of a constitutively active gene
from which an mRNA molecule can be transcribed,
and protein, the production of which depends on the
instantaneous abundance of mRNA (see Fig. 1A).
Both mRNA and protein are subjected to stochas-
tic degradation. Such a qualitative model can
be described mathematically as a two-dimensional
Markov jump process in the copy numbers of mRNA
and protein, with reaction probabilities that are func-
tions of the current state only (hence the Markov
property), and suitably chosen kinetic constants [1,
2].
While the two-stage model is easily simulated us-
ing stochastic simulation algorithms such as Gille-
spie’s algorithm [3], it is nonetheless a difficult task
to derive analytical expressions for the evolution of
mRNA and protein copy numbers with time. The
Markov process itself obeys the chemical master
equation (CME), an infinite-dimensional system of
ordinary differential equations, for which no exact
(closed-form) solutions are known in general. Nu-
merous approaches exist for the approximate solu-
tion of the CME, such as the linear noise approxi-
mation [4], a second-order Taylor series expansion
in the system size of the reaction volume; moment
equations and variants thereof [5, 6], which capture
an arbitrary number of statistical moments of the
stochastic process; finite state projection [7], a trun-
cation of the state-space of possible copy number
combinations, and many others (for an overview, see
1

Journal of Coupled Systems and Multiscale Dynamics
[8]). We further note that this particular model has
been studied using a variety of analytical and com-
putational techniques, see e.g. [912] or [13] for a
review of related modelling approaches.
An alternative analytical approach was developed
by Shahrezaei and Swain [2], wherein it is assumed
that mRNA molecules decay much faster than pro-
tein, a realistic assumption in many prokaryotic
cells. In the limit of a perfect scale separation in
which the decay of mRNA is instantaneous, the
CME underlying the two-stage model can be solved
analytically by the introduction of a generating func-
tion. The latter then obeys a first-order linear par-
tial differential equation, the solution of which can
be obtained via the method of characteristics. The
resulting analytical expression for the general time-
dependent joint probability density of mRNA and
protein, called the propagator of the system, is of
great utility for understanding its dynamics in time.
However, it is not valid when the assumption of
scale separation is violated, as is commonly the case
for eukaryotic cells. In recent work [14], the proce-
dure developed in [2] was extended to capture de-
parture from the assumption of perfect scale sepa-
ration: the ratio of degradation rates of protein and
mRNA, denoted ε, was taken to be small and pos-
itive instead of zero, as was the case in [2]. The
presence of the (singular) perturbation parameter ε
allows for the application of asymptotic techniques,
such as geometric singular perturbation theory [15]
and matched asymptotic expansions [16].
In the present case study, we explore the util-
ity of this newly developed perturbative approach
for propagator-based parameter inference in systems
with varying degrees of scale separation. Specif-
ically, our goal is to estimate molecular parame-
ters in the model from observations of protein abun-
dance only. Trajectories are simulated via Gille-
spie’s stochastic simulation algorithm in a param-
eter regime in which mRNA and protein are pro-
duced continuously, i.e., not in translational bursts.
The protein time-courses are sampled at regular time
intervals, thus mimicking a typical time-lapse flu-
orescence microscopy setup [17, 18]. While fluo-
rescence microscopy yields only time-series for the
intensity, these can nonetheless be converted into
absolute protein numbers if a calibration factor of
molecules per unit intensity can be estimated, see
e.g. [19]. We note that mRNA time-courses are not
observed, and that they are hence not used for pa-
rameter inference.
The zeroth-order propagator obtained by setting
ε = 0 [2] is then compared to a first-order propaga-
tor (in ε > 0) that is uniformly valid both on short
and on long time-scales [14], in terms of the abil-
ity of each to capture the correct parameters i.e.,
the kinetic constants in the underlying chemical re-
action network in the two-stage model for gene
expression. For comparison, both propagators are
also contrasted with an approximate solution of the
CME that is computed using a finite state projection.
A number of simplifying assumptions are made; no-
tably, we ignore impeding factors such as measure-
ment noise, uncertainty in the conversion from flu-
orescence intensity to protein numbers or low sam-
pling frequency of fluorescent signal. Rather, our
focus in this case study is on assessing the general
efficiency and accuracy of the propagator-based ap-
proach for parameter inference.
2. METHODS
2.1. Two-stage Gene Expression Model
We model gene expression as a two-stage process,
whereby DNA is transcribed to mRNA, which is
then translated into protein (see Fig. 1A). Denoting
the probability of observing m molecules of mRNA
and n molecules of protein in the system at time τ
by P
m,n
(τ), we find that the latter evolves according
to the non-dimensionalized CME [2, 4]
P
m,n
τ
= a(P
m1,n
P
m,n
)
+ γbm(P
m,n1
P
m,n
)
+ γ[(m + 1)P
m+1,n
mP
m,n
]
+ [(n + 1)P
m,n+1
nP
m,n
]. (1)
Here, m and n denote mRNA and protein copy num-
bers, respectively, a is the non-dimensional tran-
scription rate and b is the non-dimensional trans-
lation rate, while the degradation rates of mRNA
and protein are given by γ and 1, respectively
(cf. Fig. 1A). Finally, τ denotes a suitably non-
dimensionalized time variable.
As in [2, 14], we define the perturbation parame-
ter ε = γ
1
here. It follows that for ε sufficiently
small, the dynamics of Eq. (1) will vary on two
distinct time-scales: the long-term behavior of the
system is naturally described on the “slow” τ -scale,
while the “fast” transients evolve according to the
rescaled time t :=
τ
ε
.
2.2. Propagator Expressions
In this section, we collect a number of analytical
results that underly the present case study; details
can be found in [2, 14].
2

Journal of Coupled Systems and Multiscale Dynamics
2.2.1. Zeroth-Order Propagator
The zeroth-order propagator for the two-stage
gene expression model (Fig. 1A) represents an ap-
proximation to the CME, Eq. (1), under the assump-
tion of infinitely fast mRNA degradation. Math-
ematically speaking, it is obtained in the singular
limit of γ , i.e., of ε 0. Following [2],
we have
P
n|n
0
(τ, 0) = (1 e
τ
)
n
0
1 + be
τ
1 + b
a
b
1 + b
n
n
X
k=0
(1)
k
k!(n k)!
Γ(a + n k)Γ(n
0
+ 1)
Γ(a)Γ(n
0
k + 1)
×
h
1 + b
b(1 e
τ
)
i
k
2
F
1
n + k, a, 1 a n + k,
1 + b
e
τ
+ b
(2)
for the zeroth-order marginal probability P
n|n
0
(τ, 0)
of observing n protein molecules after time τ , given
m
0
= 0 molecules of mRNA and n
0
molecules of
protein initially. Here,
2
F
1
(a, b, c, z) is the Gauss
hypergeometric function [20]. We remark that, by
construction, P
n|n
0
(τ, 0) neglects any contributions
from the fast t-scale, as the decay of mRNA is in-
stantaneous to leading order in ε.
2.2.2. Uniform (First-Order) Propagator
The uniform propagator, denoted P
n|n
0
(τ, t, ε),
was derived as in [14]. Here, ε denotes the per-
turbation parameter, as before, while t is the fast
time variable. We emphasize that P
n|n
0
describes
the probability of transitioning from n
0
protein
molecules initially to n molecules at time τ = εt,
uniformly on the two time-scales. After some alge-
braic rearrangement, we find
P
n|n
0
(τ, t, ε) = P
n|n
0
(τ, ε)
+ εa
b
1 + b
nn
0
1
(1 + b)
2
× [n n
0
b (1 + b)t] +
εa
Γ(n n
0
+ 2)
(bt)
nn
0
t
×
1
F
1
(n n
0
+ 1, n n
0
+ 2, (1 + b)t)t
1
n n
0
b
1 + b
+
n n
0
+ 1
1 + b
e
(1+b)t
(3)
to first order in ε; here,
1
F
1
(a, b, z) is the Kum-
mer function of the first kind (or confluent hyper-
geometric function) [20]. We remark that the transi-
tion probability P
n|n
0
(τ, ε) contributes on the slow
τ-scale in Eq. (3), while the t-dependent contribu-
tion in Eq. (3) accounts for the transient dynamics
on the fast time-scale.
Specifically, P
n|n
0
(τ, ε) denotes the marginal
probability, up to and including O(ε)-terms, of ob-
serving n protein molecules after time τ given m
0
=
0 molecules of mRNA and n
0
molecules of protein
initially:
P
n|n
0
(τ, ε) =
X
m=0
P
m,n|0,n
0
(τ, ε) (4)
As shown in [14], the probability of encountering
more than 1 molecule of mRNA at time τ is negligi-
ble to first order in ε; thus, Eq. (4) reduces to
P
n|n
0
(τ, ε) = P
0,n|0,n
0
(τ, ε) + P
1,n|0,n
0
(τ, ε).
(5)
3

Journal of Coupled Systems and Multiscale Dynamics
After some algebraic simplification, the two tran-
sition probabilities P
0,n|0,n
0
and P
1,n|0,n
0
in the
above relation are found to be
P
0,n|0,n
0
(τ, ε) = (1 e
τ
)
n
0
b
1 + b
n
1 + be
τ
1 + b
a
×
n
X
k=0
1
(n k)B(a, n k)
2
F
1
n + k, a, 1 a n + k,
1+b
e
τ
+b
×
g(n
0
, k)
ε
2
a
(1 + b)
2
(k + 1) ×
h
2
F
1
k, n
0
, 1 k,
1+b
b(1e
τ
)
+
1 + b
e
τ
+ b
k+2
e
2τ
2
F
1
k, n
0
, 1 k,
e
τ
+b
b(1e
τ
)
i
, with
g(n
0
, k) =
(
0 for k > n
0
(1)
k
n
0
k

(1+b)
b(1e
τ
)
k
for k n
0
;
(6)
P
1,n|0,n
0
(τ, ε) =
b
b + 1
n
1
b + 1
(1 e
τ
)
n
0
1 + be
τ
1 + b
a
×
n
X
k=0
1
(n k)B(a, n k)
2
F
1
k n, a, a + k n + 1,
b+1
e
τ
+b
×
h(n
0
, k) + (1)
n
0
h
be
τ
+ 1
b(1 e
τ
)
i
n
0

, with
h(n
0
, k) =
(
0 for k n
0
n
0
k+1

b+1
b(1e
τ
)
k+1
2
F
1
1, k n
0
+ 1, k + 2,
b+1
b(1e
τ
)
for k < n
0
.
(7)
Here, B(a, b) :=
Γ(a)Γ(b)
Γ(a+b)
is the Beta function, with
the proviso that
1
(nk)B(a,nk)
= 1 when n = k.
Finally, Eq. (3) can be simplified by substituting
1
F
1
(n n
0
+ 1; n n
0
+ 2; (1 + b)t) = [(1 + b)t]
(nn
0
+1)
Γ(n n
0
+ 2)
(n n
0
+ 1)Γ
n n
0
+ 1, (1 + b)t
(8)
to achieve the computationally more tractable for-
mulation
P
n|n
0
(τ, t, ε) = P
n|n
0
(τ, ε)
+ εa
b
1 + b
nn
0
1
(1 + b)
2
[n n
0
b (1 + b)t] + εat
b
1 + b
nn
0
1
(1 + b)t
×
b + n
0
n
1 + b
t
1 Q(n n
0
+ 1, (1 + b)t)
+
(bt)
(nn
0
)
1 + b
e
(1+b)t
Γ(n n
0
+ 1)
. (9)
4

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Citations
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Journal ArticleDOI
TL;DR: The classical two-state model of stochastic mRNA dynamics in eukaryotic cells is extended to include a considerable number of salient features of single-cell biology, such as cell division, replication, mRNA maturation, dosage compensation, and growth-dependent transcription, and derive expressions for the approximate time-dependent protein-number distributions.
Abstract: The stochasticity of gene expression presents significant challenges to the modeling of genetic networks. A two-state model describing promoter switching, transcription, and messenger RNA (mRNA) decay is the standard model of stochastic mRNA dynamics in eukaryotic cells. Here, we extend this model to include mRNA maturation, cell division, gene replication, dosage compensation, and growth-dependent transcription. We derive expressions for the time-dependent distributions of nascent mRNA and mature mRNA numbers, provided two assumptions hold: 1) nascent mRNA dynamics are much faster than those of mature mRNA; and 2) gene-inactivation events occur far more frequently than gene-activation events. We confirm that thousands of eukaryotic genes satisfy these assumptions by using data from yeast, mouse, and human cells. We use the expressions to perform a sensitivity analysis of the coefficient of variation of mRNA fluctuations averaged over the cell cycle, for a large number of genes in mouse embryonic stem cells, identifying degradation and gene-activation rates as the most sensitive parameters. Furthermore, it is shown that, despite the model's complexity, the time-dependent distributions predicted by our model are generally well approximated by the negative binomial distribution. Finally, we extend our model to include translation, protein decay, and auto-regulatory feedback, and derive expressions for the approximate time-dependent protein-number distributions, assuming slow protein decay. Our expressions enable us to study how complex biological processes contribute to the fluctuations of gene products in eukaryotic cells, as well as allowing a detailed quantitative comparison with experimental data via maximum-likelihood methods.

51 citations


Journal ArticleDOI
TL;DR: An analytical method is proposed for the efficient approximation of propagators of stochastic models for gene expression which lends itself naturally to implementation in a Bayesian parameter inference scheme, and can be generalised systematically to related categories of stoChastic models beyond the ones considered here.
Abstract: The inherent stochasticity of gene expression in the context of regulatory networks profoundly influences the dynamics of the involved species. Mathematically speaking, the propagators which describe the evolution of such networks in time are typically defined as solutions of the corresponding chemical master equation (CME). However, it is not possible in general to obtain exact solutions to the CME in closed form, which is due largely to its high dimensionality. In the present article, we propose an analytical method for the efficient approximation of these propagators. We illustrate our method on the basis of two categories of stochastic models for gene expression that have been discussed in the literature. The requisite procedure consists of three steps: a probability-generating function is introduced which transforms the CME into (a system of) partial differential equations (PDEs); application of the method of characteristics then yields (a system of) ordinary differential equations (ODEs) which can be solved using dynamical systems techniques, giving closed-form expressions for the generating function; finally, propagator probabilities can be reconstructed numerically from these expressions via the Cauchy integral formula. The resulting 'library' of propagators lends itself naturally to implementation in a Bayesian parameter inference scheme, and can be generalised systematically to related categories of stochastic models beyond the ones considered here.

21 citations


Cites background or methods from "A case study on the use of scale se..."

  • ...An example realisation of the above procedure can e.g. be found in the work by Feigelman et al. (2015)....

    [...]

  • ...…be applied in a straightforward fashion in cases where Assumption 3.2 fails, which is particularly relevant in relation to previous work (Shahrezaei and Swain 2008a; Feigelman et al. 2015), where the CME system (3.3) is studied for parameter values far beyond the range implied by Assumption 3.2....

    [...]


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15 Dec 2020-bioRxiv
TL;DR: An artificial neural network is used to approximate the time-dependent distributions of non-Markov models by the solutions of much simpler time-inhomogeneous Markov models; the approximation does not increase the dimensionality of the model and simultaneously leads to inference of the kinetic parameters.
Abstract: Non-Markov models of stochastic biochemical kinetics often incorporate explicit time delays to effectively model large numbers of intermediate biochemical processes. Analysis and simulation of these models, as well as the inference of their parameters from data, are fraught with difficulties because the dynamics depends on the system’s history. Here we use an artificial neural network to approximate the time-dependent distributions of non-Markov models by the solutions of much simpler time-inhomogeneous Markov models; the approximation does not increase the dimensionality of the model and simultaneously leads to inference of the kinetic parameters. The training of the neural network uses a relatively small set of noisy measurements generated by experimental data or stochastic simulations of the non-Markov model. We show using a variety of models, where the delays stem from transcriptional processes and feedback control, that the Markov models learnt by the neural network accurately reflect the stochastic dynamics across parameter space.

7 citations


Posted ContentDOI
25 Oct 2020-bioRxiv
TL;DR: It is found that the solution of the chemical master equation – including static extrinsic noise – exactly agrees with the agent-based formulation when the network under study exhibits stochastic concentration homeostasis, a novel condition that generalises concentrationHomeostasis in deterministic systems to higher order moments and distributions.
Abstract: The chemical master equation and the stochastic simulation algorithm are widely used to model the reaction kinetics inside living cells. It is thereby assumed that cell growth and division can be modelled for through effective dilution reactions and extrinsic noise sources. We here re-examine these paradigms through developing an analytical agent-based framework of growing and dividing cells accompanied by an exact simulation algorithm, which allows us to quantify the dynamics of virtually any intracellular reaction network affected by stochastic cell size control and division noise in a growing population. We find that the solution of the chemical master equation – including static extrinsic noise – exactly agrees with the one of the agent-based formulation when a simple condition on the network’s topology is met. We illustrate this result for a range of common gene expression networks. When these conditions are not met, we demonstrate using analytical solutions of the agent-based models that the dependence of gene expression noise on cell size can qualitatively deviate from the effective master equation. Surprisingly, the latter distorts total noise in gene regulatory networks by at most 8% independently of network parameters. Our results highlight the accuracy of extrinsic noise modelling within the chemical master equation framework.

2 citations


Journal ArticleDOI
Abstract: Stochastic gene expression in regulatory networks is conventionally modelled via the Chemical Master Equation (CME) (van Kampen 1981). As explicit solutions to the CME, in the form of so-called propagators, are not readily available, various approximations have been proposed (Zechner et al. 2013, Feigelman et al 2016, Popovic, Marr and Swain 2016). A recently developed analytical method (Veerman, Marr and Popovic 2017) is based on a scale separation that assumes significant differences in the lifetimes of mRNA and protein in the network, allowing for the efficient approximation of propagators from asymptotic expansions for the corresponding generating functions. Here, we showcase the applicability of that method to a ‘telegraph’ model for gene expression that is extended with an autoregulatory mechanism. We demonstrate that the resulting approximate propagators can be successfully applied for Bayesian parameter inference in the non-regulated model with synthetic data; moreover, we show that in the extended autoregulated model, autoactivation or autorepression may be refuted under certain assumptions on the model parameters. Our results indicate that the method showcased here may allow for successful parameter inference and model identification from longitudinal single cell data.

2 citations


References
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01 Jan 2016
TL;DR: The handbook of mathematical functions is universally compatible with any devices to read and is available in the digital library an online access to it is set as public so you can get it instantly.
Abstract: Thank you for reading handbook of mathematical functions. As you may know, people have look numerous times for their favorite readings like this handbook of mathematical functions, but end up in infectious downloads. Rather than enjoying a good book with a cup of tea in the afternoon, instead they are facing with some infectious bugs inside their laptop. handbook of mathematical functions is available in our digital library an online access to it is set as public so you can get it instantly. Our digital library spans in multiple countries, allowing you to get the most less latency time to download any of our books like this one. Kindly say, the handbook of mathematical functions is universally compatible with any devices to read.

2,181 citations


Book
01 Jan 2005
TL;DR: The Simplex Method for Linear Programming Problems is a method for solving linear programming problems with real-time requirements.
Abstract: Preface Table of Notation Chapter 1. Introduction Chapter 2. Line Search Descent Methods for Unconstrained Minimization Chapter 3. Standard Methods for Constrained Optimization Chapter 4. New Gradient-Based Trajectory and Approximation Methods Chapter 5. Example Problems Chapter 6. Some Theorems Chapter 7. The Simplex Method for Linear Programming Problems Bibliography Index

799 citations


Book ChapterDOI
01 Jan 2013
Abstract: The ideas underlying an asymptotic approximation appeared in the early 1800s when there was considerable interest in developing formulas to evaluate special functions. An example is the expansion of Bessel’s function, given in ( 1.15), that was derived by Poisson in 1823. It was not until later in the century that the concept of an asymptotic solution of a differential equation took form, and the most significant efforts in this direction were connected with celestial mechanics. The subject of this chapter, what is traditionally known as matched asymptotic expansions, appeared somewhat later. Its early history is strongly associated with fluid mechanics and, specifically, aerodynamics. The initial development of the subject is credited to Prandtl (1905), who was concerned with the flow of a fluid past a solid body (such as an airplane wing). The partial differential equations for viscous fluid flow are quite complicated, but he argued that under certain conditions the effects of viscosity are concentrated in a narrow layer near the surface of the body. This happens, for example, with air flow across an airplane wing, and a picture of this situation is shown in Fig. 2.1. This observation allowed Prandtl to go through an order-of-magnitude argument and omit terms he felt to be negligible in the equations. The result was a problem that he was able to solve. This was a brilliant piece of work, but it relied strongly on his physical intuition. For this reason there were numerous questions about his reduction that went unresolved for decades. For example, it was unclear how to obtain the correction to his approximation, and it is now thought that Prandtl’s derivation of the second term is incorrect (Lagerstrom, 1988). This predicament was resolved when Friedrichs (1941) was able to show how to systematically reduce a boundary-layer problem. In analyzing a model problem (Exercise 2.1) he used a stretching transformation to match inner and outer solutions, which is the basis of the method that is discussed in this chapter. This procedure was not new, however, as demonstrated by the way in which Gans (1915) used some of these ideas to solve problems in optics.

17 citations


"A case study on the use of scale se..." refers methods in this paper

  • ...The presence of the (singular) perturbation parameter ε allows for the application of asymptotic techniques, such as geometric singular perturbation theory [15] and matched asymptotic expansions [16]....

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